Systems and methods of coordination control for robot manipulation

ABSTRACT

Disclosed herein are systems and methods for controlling robotic apparatus having several movable elements or segments coupled by joints. At least one of the movable elements can include one or more mobile bases, while the others can form one or more manipulators. One of the movable elements can be treated as an end effector for which a certain motion is desired. The end effector may include a tool, for example, or represent a robotic hand (or a point thereon), or one or more of the one or more mobile bases. In accordance with the systems and methods disclosed herein, movement of the manipulator and the mobile base can be controlled and coordinated to effect a desired motion for the end effector. In many cases, the motion can include simultaneously moving the manipulator and the mobile base.

REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of priority of U.S. ProvisionalApplication No. 61/115,461, titled “SYSTEM AND METHOD OF COORDINATIONCONTROL FOR ROBOT MANIPULATION” and filed Nov. 17, 2008, and also claimsthe benefit of priority of U.S. Provisional Application No. 61/115,468,titled “HUMAN-ROBOT MANIPULATION FOR COMPLEX OPERATIONS” and filed Nov.17, 2008, both of which are hereby incorporated by reference in itsentirety.

FEDERALLY SPONSORED RESEARCH AND DEVELOPMENT

This invention was made with Government support under contract no.N66001-07-M-1079 with the Space & Naval Warfare Systems Center of theU.S. Navy and contract NNC06CA27C with NASA Jet Propulsion Laboratory.The U.S. Government has certain rights in the invention.

FIELD

The present application generally relates to systems and methods forcontrolling robotic apparatus such as, for example, manipulators, arms,and mobile bases.

BACKGROUND

Robotic apparatus, e.g., of the type making up remote autonomous agents,often act (or execute instructions) according to control softwareimplementing one or more control algorithms. Those control algorithmstransform input from, e.g., a user or sensors, into robotic action.

Robotic systems are increasingly called upon to perform ever morecomplicated types of tasks. Such tasks often require complexcoordination amongst various elements of a robotic system. However,coordinating control amongst several robotic elements (e.g., a movingarm, tool attached to the arm, mobile base, legs, and so on) has provendifficult and remains computationally intensive. Of particulardifficulty is the coordination of simultaneous control (e.g., to effectsimultaneous motion) of one or more arms with one or more mobile basesbased on a desired motion of, for example, one element of the system.Further, user interfaces for controlling such robotic apparatus can belimited in the information they provide to a user as the user attemptsto direct the action of the robot.

Accordingly, there is a need for improved systems and methods forcoordinating control of various elements making up a robotic apparatus,and for improved user interfaces for controlling robotic apparatus.

SUMMARY

The present application is generally directed to methods and systems forcoordinated control of robotic systems, including coordinated control ofrobotic systems that include a mobile base and an articulating arm. Inmany cases, the systems and methods of the invention allow forsimultaneous movement of the base and the arm so as to transition arobot from an initial state (e.g., characterized by an initial positionand pose or angular orientation) to a desired final state. In someimplementations the base and the arm are modeled as a plurality of linksor elements. A control vector is formed by concatenating componentscorresponding to the time rate of change of position of the linkassociated with the base (e.g., linear and angular velocities of thebase) with the respective components corresponding to other links thatare associated, e.g., with the arm. The optimal values of the componentsof the control vector(s) for motion in the next and subsequent timeintervals can be determined by optimizing a merit function, e.g., whichcan be a scalar function in many implementations. Control signals basedon the optimized values of the control vector components are applied tothe base and one or more elements of the arm to cause motion. In manycases, the motion of the base and one or more elements of the arm isconcurrent.

In one aspect, a method is disclosed for controlling a robotic apparatuswhich includes a plurality of movable elements coupled to one anothervia one or more joints defining one or more degrees of freedom ofmovement for each of the movable elements, at least one of the movableelements comprising a mobile base for the apparatus. The control methodcan include receiving input signals indicating a desired position ormotion constraint for at least one of the movable elements, andgenerating, based on the input signals, a plurality of control signalsto control movement of the mobile base and at least one other movableelement to achieve the desired motion by optimizing a pre-defined scalarfunction of the motion and positions of the movable elements and themobile base. The method further includes applying the control signals tothe mobile base and the at least one other movable element toconcurrently move the mobile base and the at least one other movableelement to effect the desired motion. In some cases, the method caninclude applying the control signals to one or more actuators coupled tothe one or more movable elements to effect the desired motion.

In some embodiments, the plurality of movable elements included in therobotic apparatus can form or include a robotic arm(s) or mobilebase(s). In some embodiments, at least one of the movable elementsincludes or represents an end effector. The end effector can be a toolsuch as a gripper, a cutter, a saw, a haptic element, or amulti-fingered end effector. In many embodiments, at least one of themovable elements (in many cases, the mobile base) provides contact withthe environment (e.g., the external environment) for the purpose oflocomotion. The mobile base can include any of a wheel, a track, and arail, or in other embodiments, any of a digging, climbing, andspelunking apparatus, and in other embodiments, any of a boat,submarine, and hovercraft, and in yet other embodiments, any of anairplane, helicopter, and lighter-than-air vehicle. The mobile base canhave two or more degrees of freedom of movement comprising linearvelocity and angular velocity. In some cases the mobile base can beconstrained in its movement, e.g., for modeling a device operating on atrack, for example, which may not be able to move laterally. In somecases, the movable elements of the robotic apparatus exhibit,collectively, at least seven degrees of freedom of movement.

In some embodiments, the input signals can be generated by a user inputdevice, and/or can be over any of audio signal, electromagnetic signal,or a computer network. In some embodiments, the method can includedisplaying (e.g., via a user display such as a computer screen, and insome cases simultaneously displaying) a plurality of views of therobotic apparatus pose, and allowing the user to control the roboticapparatus from each of the plurality of views with the user inputdevice. As one skilled in the art will understand, other views, to whichthe user cannot switch to effect control purposes, can additionally beincluded. The user input device can require only a single user action toswitch control between the plurality of views. The single user actioncan include, for example, a mouseclick, a mouseover, a selection from amenu, a touch on a touch sensitive screen, a depression of a key orother tactile sensor, or a command-line entry (e.g., in a text-basedinterface). In some embodiments, the method can include displaying aplurality of views of the robotic apparatus pose, allowing a user tocontrol the robotic apparatus from each of the plurality of views, andautomatically updating each of plurality of the views based on a user'scontrol input in any one view. In some embodiments, the plurality ofviews can include rendered views.

In some embodiments, the scalar function (also referred to as a meritfunction herein) can be associated with any of: kinetic energyassociated with movement of the robotic apparatus, a gradient ofpotential energy associated with movement of the robotic apparatus,singularity avoidance, accuracy, joint-limit avoidance, obstacleavoidance. The scalar function can signify a constraint on movement ofat least one movable element. In some cases, the scalar function can bedefined by:

$\mu = {{\frac{1}{2}{\overset{.}{q}}^{T}W\overset{.}{q}} + {\alpha {\overset{.}{q}}^{T}{F.}}}$

where q is a n-length vector representation of the positions of the oneor more joints and the mobile base; with {dot over (q)} being its timederivative (change in position over time); W is a matrix function W(q);F is a vector function of configuration; and α is a scalar function ofconfiguration. The parameters W, F and α can be selected to define acontrol strategy for controlling the robotic apparatus.

In yet other embodiments, the generation of control signals in thecontrol method can include determining control vectors (e.g., {dot over(q)}) using the following equation:

$\overset{.}{q} = {\left\lbrack \frac{J}{N_{J}^{T}W} \right\rbrack^{- 1}\left\lbrack \frac{V}{{- \alpha}\; N_{J}^{T}F} \right\rbrack}$

where q is a n-length vector representation of the positions of the oneor more joints and the mobile base; with {dot over (q)} being its timederivative (change in position over time); V is a m-length vectorrepresentation of desired motion of at least one of the plurality ofmovable elements; J is an (m×n) manipulator Jacobian; N_(J) ^(T) is ann×(n−m) set of vectors defined such that JN_(J) ^(T)=0; W is a matrixfunction of q; F is a vector function of q; and α is a scalar functionof q. In some embodiments, V can be specified (e.g., by a user) forseveral of, the majority of, or all of the plurality of movableelements.

In another aspect, a robotic system is disclosed which includes aplurality of movable elements coupled to one another via one or morejoints defining one or more degrees of freedom of movement for each ofthe movable elements, at least one of said movable elements comprising amobile base. The robotic system can also include a user input device forreceiving input signals indicating a desired position or motionconstraint for at least one point on one of the movable elements, andone or more processors coupled to the input device and generating aplurality of control signals to control movement of the mobile base andat least one other movable element to achieve the desired motion byoptimizing a pre-defined scalar function of motion and positions of themovable elements and the mobile base. The robotic system can furtherinclude one or more actuators coupled to the plurality of movableelements and to the one or more processors, the one or more actuatorseffecting the desired motion by concurrently moving the mobile base andthe at least one other movable element in accordance with the controlsignals.

A wide array of variations are possible. For example, in one embodiment,at least one of the movable elements can comprise an end effector whichincludes any of a gripper, a cutter, a saw, a haptic element, amulti-fingered end effector. In some embodiments, the base can includeany of a wheel, a track, and a rail. In some embodiments, the pluralityof movable elements comprise a robotic arm and a mobile base.

In yet further embodiments, the scalar function can be defined by:

$\mu = {{\frac{1}{2}{\overset{.}{q}}^{T}W\overset{.}{q}} + {\alpha {\overset{.}{q}}^{T}{F.}}}$

where q is a n-length vector representation of the positions of the oneor more joints and the mobile base; with {dot over (q)} being its timederivative (change in position over time); W is a matrix function W(q);F is a vector function of configuration; and α is a scalar function ofconfiguration.

In some embodiments, the system can further include an operator controlunit that (i) displays a plurality of views of the robotic apparatuspose (e.g., via a user display), and (ii) allows a user to control therobotic apparatus from each of the plurality of the views (e.g., via theaforementioned user input device, which can be incorporated into or partof the operator control unit). The operator control unit (e.g., via itssoftware user interface) can require only a single user action to switchcontrol between the plurality of views. In some embodiments, the systemcan include an operator control unit that (i) displays a plurality ofviews of the robotic apparatus pose, (ii) allows a user to control therobotic apparatus from each of the plurality of views, and (iii)automatically updates each of the plurality of views based on a user'scontrol input in any one view. In some embodiments, the plurality ofviews can include a plurality of rendered views.

In another aspect, a robotic system is disclosed which includes aplurality of movable elements coupled to one another via one or morejoints defining one or more degrees of freedom of movement for each ofthe movable elements, at least one of said movable elements comprising amobile base. The system can further include a user input device forreceiving input signals indicating a desired motion for at least onepoint on one of the movable elements, and one or more processorsreceiving the input signals and generating a plurality of controlsignals to control movement of the mobile base and at least one othermovable element to achieve the desired motion. Further, the system caninclude one or more actuators coupled to the plurality of movableelements and to the one or more processors, the one or more actuatorseffecting the desired motion by concurrently moving the mobile base andthe at least one other movable element in accordance with the controlsignals. In many embodiments, the one or more processors can generatethe control signals using the following equation:

$\overset{.}{q} = {\left\lbrack \frac{J}{N_{J}^{T}W} \right\rbrack^{- 1}\left\lbrack \frac{V}{{- \alpha}\; N_{J}^{T}F} \right\rbrack}$

where q is a n-length vector representation of the positions of the oneor more joints and the mobile base; with {dot over (q)} being its timederivative; V is a m-length vector representation of desired motion ofat least one point of one of the plurality of movable elements; J is an(m×n) manipulator Jacobian; N_(J) ^(T) is an n×(n−m) set of vectorsdefined such that JN_(J) ^(T)=0; W is a matrix function W(q); F is thegradient of scalar function f(q); α is a scalar. In some embodiments, Vcan be specified (e.g., by a user) for several of, the majority of, orall of the plurality of movable elements. In many cases, the parametersW, F and α are user-selected to define a control strategy forcontrolling the robotic apparatus. Further, in some embodiments, the oneor more processor can generate the control signals so as to optimize amerit function defined as:

$\mu = {{\frac{1}{2}{\overset{.}{q}}^{T}W\overset{.}{q}} + {{\overset{.}{q}}^{T}F\; \alpha}}$

where μ represents a parameter to be optimized that reflects the controlstrategy.

In some embodiments, the system can further include an operator controlunit that (i) displays a plurality of views of the robotic apparatuspose (e.g., via a user display), and (ii) allows a user to control therobotic apparatus from each of the plurality of views (e.g., via a userinput device). The operator control unit (e.g., via its software userinterface) can require only a single user action to switch controlbetween a plurality of the views. Further, in some embodiments, thesystem can include an operator control unit that (i) displays aplurality of views of the robotic apparatus pose, (ii) allows a user tocontrol the robotic apparatus from each of the plurality of the views,and (iii) automatically updates each of the plurality of views based ona user's control input in any one view. In some embodiments, theplurality of views can include a plurality of rendered views.

In another aspect, a method is disclosed for controlling a roboticapparatus which includes a plurality of movable elements coupled to oneanother via one or more joints defining one or more degrees of freedomof movement (and in some cases at least seven degrees of freedom ofmovement, and in other cases more than seven) for each of the movableelements, with at least one of the movable elements comprising a mobilebase for the apparatus. The control method can include input signalsindicating a desired motion for at least one point on one of the movableelements, which can be treated as an end effector element. The methodcan further include determining, based on the desired motion of the atleast one point on the end effector element, a plurality of controlvectors representing position changes for the one or more joints and themobile base to achieve the desired motion. The control vectors canrepresent time rate of position change, for example, and they can bedetermined so as to optimize a merit function associated with auser-desired control strategy. The control vectors are applied to themobile base and the at least one other movable element to concurrentlymove the mobile base and the at least one other movable element andeffect the desired motion.

In some embodiments, the method can further include modeling theplurality of movable elements as a plurality of links, at least one linkmodeling the mobile base (“base link”), at least one link including atleast one point for which a desired motion is sought (“end effectorlink”), and at least one link representing an aspect link, the movementof other links being determined in relation to the aspect link.

In another aspect, a method is disclosed for controlling a roboticapparatus which includes a plurality of movable elements coupled to oneanother via one or more joints defining one or more degrees of freedomof movement for each of the movable elements, at least one of saidmovable elements comprising a mobile base for the apparatus. The controlmethod can include receiving input signals indicating a desired motionfor at least one point on one of the movable elements, and calculating,with a digital data processor, a plurality of control signals to controlmovement of the mobile base and at least one other movable element.Further, those control signals can be applied to the mobile base and theat least one other movable element to concurrently move the mobile baseand the at least one other movable element and effect the desiredmotion. The calculation of the control signals can use the equation:

$\overset{.}{q} = {\left\lbrack \frac{J}{N_{J}^{T}W} \right\rbrack^{- 1}\left\lbrack \frac{V}{{- \alpha}\; N_{J}^{T}F} \right\rbrack}$

where q is a n-length vector representation of the positions of the oneor more joints and the mobile base; with {dot over (q)} being its timederivative; V is a m-length vector representation of a desired motion ofthe plurality of movable elements; J is an (m×n) manipulator Jacobian;N_(J) ^(T) is an n×(n−m) set of vectors defined such that JN_(J) ^(T)=0;W is a matrix function W(q); F is the gradient of scalar function f(q);α is a scalar. The foregoing equation can optimize the following meritfunction:

$\mu = {{\frac{1}{2}{\overset{.}{q}}^{T}W\overset{.}{q}} + {{\overset{.}{q}}^{T}F\; \alpha}}$

where μ represents a parameter to be optimized that reflects a controlstrategy.

In another aspect, a robot is disclosed which includes a mobile basecapable of moving in at least one dimension, an articulating arm coupledto the mobile base, and a control system for generating control signalsfor application to the base and the arm so as to move the base whileconcurrently changing a configuration of the arm. The control system canbe configured to determine values of a plurality of velocity vectorsrepresenting time-derivative of a plurality of vectors indicative ofpositions of the base and a configuration of the arm and generating thecontrol signals based on the velocity vectors. The control system candetermine those velocity vector values by optimizing a merit function,which may incorporate one or more constraints associated with motion ofthe base and the arm.

In yet another aspect, a robotic system is disclosed which includes aplurality of movable elements coupled to one another via one or morejoints defining one or more degrees of freedom of movement for each ofthe movable elements, at least one of said movable elements comprising amobile base. The system can further include a user input device forreceiving input signals indicating a desired motion for at least onepoint on one of the movable elements (“end effector element”), and oneor more processors communicatively coupled to the input device andgenerating, based on the input signals, control vector data representingposition changes for the one or more joints and the mobile base toachieve the desired motion. One or more actuators can be coupled to theplurality of movable elements and to the one or more processors, the oneor more actuators effecting the desired motion by concurrently movingthe mobile base and the at least one other movable element in accordancewith the control vectors.

BRIEF DESCRIPTION OF THE DRAWINGS

A further understanding of various aspects of the systems and methodsdiscussed herein can be obtained by reference to the following detaileddescription in conjunction with the associated drawings, in which:

FIG. 1 schematically depicts an exemplary robotic apparatus with amanipulator and a mobile base;

FIG. 2 shows a graph of an exemplary joint-limit avoidance function foruse in connection with optimizing a merit function;

FIG. 3 shows an exemplary illustration of a model for measuring thestatistical error of a Euclidean space for use in connection withoptimizing a merit function;

FIG. 4 shows an exemplary representation of several reference frames fora coordinated control model;

FIG. 5 shows an exemplary representation of several links for acoordinated control model;

FIG. 6 shows an exemplary representation of an aspect link in the modelrepresented by FIG. 5;

FIG. 7 shows an exemplary representation of an end effector link in themodel represented by FIG. 5;

FIG. 8 shows an exemplary representation of relative links in the modelrepresented by FIG. 5;

FIG. 9 shows an exemplary block diagram of a control system inaccordance with the teachings herein;

FIG. 10 shows an exemplary block diagram of a software architecture foruse with the control system of FIG. 9;

FIG. 11 shows an exemplary flow diagram of steps to be executed bydigital data processor(s) of the control system shown in FIG. 9;

FIG. 12 is a schematic depiction of an exemplary two-wheel model forsimulating a mobile base;

FIG. 13 shows an exemplary dialog box illustrating the selection of endeffector types and parameters;

FIG. 14 shows an exemplary free-floating mechanism in simulation;

FIGS. 15A-15D show a sequence of images of a exemplary model infree-base mode simulation;

FIG. 16 shows an exemplary model for joint torque which accounts forbase acceleration;

FIG. 17 shows an exemplary illustration of an articulated bodycomprising a plurality of links; and,

FIGS. 18A-18D show a side-by-side comparison of different algorithmsused to simulate the movement of free-base mechanisms.

DETAILED DESCRIPTION

Certain exemplary embodiments will now be described to provide anoverall understanding of the principles of the structure, function,manufacture, and use of the devices and methods disclosed herein. One ormore examples of these embodiments are illustrated in the accompanyingdrawings. Those skilled in the art will understand that the devices andmethods specifically described herein and illustrated in theaccompanying drawings are non-limiting exemplary embodiments and do notlimit the scope of the claims.

A variety of exemplary embodiments will be disclosed herein. Thefeatures illustrated or described in connection with one exemplaryembodiment may be combined with the features of other embodiments. Suchmodifications and variations are intended to be included within thescope of the present application.

In many cases, the methods and systems disclosed herein can beimplemented using, amongst other things, software created as directed bythe teachings herein and in accordance with an object orientedprogramming scheme. Principles of object-oriented programming andprogramming languages, (e.g., C++) are known in the art. Examples ofsoftware toolkits which can be used to implement the systems and methodsdisclosed herein include the Actin and Selectin software toolkits,developed by Energid Technologies, Cambridge, Mass., USA. Reference willbe made to these software toolkits (e.g., to certain classes availabletherein) throughout this application for illustrative purposes.

A. ROBOTIC APPARATUS WITH BASE

Generally, disclosed herein are systems and methods for controllingapparatus, such as robotic apparatus or any other device requiringcontrol. Such systems and methods include in particular those havingrobotic manipulators (e.g., an articulating arm with a tool such asgripper, grasper, saw, other tool, or a hand-like extremity) with abase. Although the base can be stationary, in many cases the base can becapable of moving (e.g., via wheels, tracks, or other mechanism). Manyof the systems and methods disclosed herein can coordinate the controland corresponding movement of a manipulator and a mobile base. Thatcoordinated control can be simultaneously effected across both themanipulator(s) and the base, e.g., both the manipulator and the base canbe simultaneously or concurrently moved to effect to a desired motion,such a user-specified motion for an end effector, of a particular partof the robotic apparatus.

FIG. 1 schematically depicts a robotic apparatus 100 made up of severalmovable elements 102, 104, 106, 108, 114. In this embodiment, theelements are coupled to one another via joints 116, 118, 120, 122 so asto create an articulating arm or manipulator 110 with a base 114. Thebase is mobile, as evidenced in FIG. 1 by several wheels 126 coupled toits sides. Virtually any other structure for moving the base could beused, such as tracks, treads, rails, cogs, propellers, jets, etc.

In this case, each of the elements 102, 104, 106, 108, 114 has one ormore degrees of freedom in which it can move. Some elements may be ableto move up and down (e.g., in a z-direction), side to side (e.g., in ay-direction), or extend distally (e.g., in an x-direction) or rotate,based for example on the capabilities of the joints coupling theelements to one another. Certain elements may be constrained in theirmovement, for example the base 114 can be on tracks limiting itssideways movement. Although any element could be so equipped, hereelement 102 includes a tool 112. In many cases such a tool 112, or theelement 102, will be treated as an end effector, for which a particularmotion is desired, however for controlling motion any element, includingthe base, can be treated as an end effector.

B. CONTROL MODEL AND CONTROL STRATEGY

In one embodiment, a robotic apparatus, such as that shown in FIG. 1,can be modeled using a framework based on the manipulator Jacobianequation:

V=J(q){dot over (q)}  (1)

where V is an m-length vector representation of the motion of one ormore end effectors, which can be any movable element(s) (usually somecombination of linear and angular velocity referenced to one or morepoints rigidly attached to parts of the robotic manipulator or thebase); q is the n-length vector of joint and base positions (with {dotover (q)} being its time derivative); and J is the m×n manipulatorJacobian, a function of q. (For spatial arms with a single end effector,V is often the frame velocity with three linear and three angularcomponents. It can also take on a larger meaning that includes theconcatenation of point, frame, or other motion of multipleend-effectors.) When the manipulator is kinematically redundant, thedimension of V is less than the dimension of q (m<n), and equation (1)is underconstrained when V is specified. By using V to representrelative motion, equation (1) can support self-connecting mechanisms bysetting the relative motion to zero.

As shown in FIG. 1, element 102 can be designated as an end effectorhaving at least one point with desired motion V.

The manipulator Jacobian specifies the rate of motion of the robot partsdoing work as a function of the rates of the degrees of freedom (jointsand base motion). By using rates rather than positions, a difficultnonlinear system description is converted into a tractable linearequation for control.

The velocity control question is the following: given a desired motionV, what are the rates {dot over (q)} that best achieve this motion? Toanswer this, the framework uses a scalar α, a matrix function W(q), anda scalar function ƒ(q) to solve for {dot over (q)}, given V, through thefollowing formula:

$\begin{matrix}{{\overset{.}{q} = {\left\lbrack \frac{J}{N_{J}^{T}W} \right\rbrack^{- 1}\left\lbrack \frac{V}{{- \alpha}\; N_{J}^{T}F} \right\rbrack}},} & (2)\end{matrix}$

where F=∇ƒ is the gradient of ƒ and N_(J) is an n×(n−m) set of vectorsthat spans the null space of J. That is, JN_(J)=0, and N_(J) has rank(n−m). Both ∇ƒ and N_(J) are generally functions of q. By changing thevalues of α, W, and ƒ, many velocity-control techniques can beimplemented.

Equation (2) optimizes the following function:

$\begin{matrix}{\mu = {{\frac{1}{2}{\overset{.}{q}}^{T}W\overset{.}{q}} + {{\overset{.}{q}}^{T}{F.}}}} & (3)\end{matrix}$

Because q represents joint positions and base positions, this controlapproach coordinates arm and base motion—it works with wheeled, tracked,and walking mobile robots, as well as snake robots.

Equation (3) can represent a merit function whose meaning depends on theselected values of α, W, and ƒ. For example, α, W, and ƒ can be selectedsuch that equation (3) represents, and equation (2) thus optimizes,kinetic energy use, avoidance of kinematic singularities, and so on.Hence, a desired control strategy (e.g., a user desired controlstrategy) can be implemented via selection of α, W, and ƒ.

Exemplary Control Strategies

A variety of control strategies are possible. The control system isflexible and able to implement a wide variety of algorithms. Forexample, control types can include, without limitation, singularityavoidance, torque minimization, obstacle avoidance, fault tolerance,minimum-kinetic-energy control, minimum-potential-energy control,accuracy optimization, and joint-limit avoidance

Singularity Avoidance

A finite-differencing tool was developed for implementing a controlstrategy that includes singularity avoidance. This tool takes a pointerto function object, and uses this function to numerically calculate itsgradient with respect to joint variables using the finite differencemethod. This is a very flexible and powerful, though somewhat costlyapproach. Generally, it is better to explicitly calculate the gradientwhenever possible. Since the explicit calculation is not alwayspossible, finite differencing can be used for singularity avoidance.

A manipulator experiences a kinematic singularity whenever the Jacobianloses rank. To frame the desire to avoid singularities into a solutionmethod, a function is needed that is large at singularities and smallaway from singularities. For this, in some embodiments, the dampedinverse of the product of the singular values of a weighted Jacobian isused. That is, the optimization function is the following:

$\begin{matrix}{{f(q)} = {\frac{1}{{{\overset{\_}{\sigma}}_{1}{\overset{\_}{\sigma}}_{2}\mspace{14mu} \ldots \mspace{14mu} {\overset{\_}{\sigma}}_{n}} + ɛ}.}} & (4)\end{matrix}$

where ε is a damping factor and σ _(i) is singular value i of theweighted Jacobian J_(W):

J_(W)=D_(T)JD_(J)  (5)

where D_(T) and D_(J) are diagonal matrices. The weighting matrices candefine the singular values of the Jacobian.

In many cases, the singular values are not explicitly calculated, asthis is too costly. Instead the Cholesky decomposition of J_(W)J_(W)^(T) is taken, and the product of the diagonal terms is used. Theproduct of these diagonal terms equals the product of the singularvalues.

In some embodiments, the gradient of the function given in equation (4)is numerically calculated for use with the basic core through the classEcControlExpressionSingularityAvoidance, which is part of theaforementioned Actin toolkit. For use with the AB core, it can becombined with EcControlExpressionVectorToAB. These can be used with apositive definite weighting matrix and a positive scalar to drive themanipulator to minimize the function and therefore move away fromkinematic singularities.

Torque Minimization

For torque minimization, the function to be optimized can be defined asthe following:

ƒ(q)g_(i) ²(q),  (6)

the square of the gravitational torque or force on joint i. To calculatethe gradient, in this exemplary embodiment an explicit gradientcalculation method (i.e., finite differencing is not used for this) forgravitational joint torque/force based on composite rigid body inertiais used. Using this calculation method, the gradient of the torquesquared is given by:

∇ƒ(q)=2g _(i)(q)∇g _(i)(q).  (7)

This is implemented in classEcControlExpressionGravitationalTorqueGradient for use with the basiccore. For use with the AB core, it can be combined withEcControlExpressionVectorToAB. Using this value as the vector input tothe core control system with a positive definite weighting matrix and apositive scalar produces control that minimized the magnitude of thegravitational torque on joint i. This can be used to prevent stress on aparticular joint, or it can be used for failure recovery after afree-swinging failure. For more information, see “Fault Tolerance forKinematically Redundant Manipulators: Anticipating Free-Swinging JointFailures,” J. D. English and A. A. Maciejewski, IEEE Transactions onRobotics and Automation, vol. 14, pp. 566-575, 1998, which is herebyincorporated by reference.

Minimum Kinetic Energy Control

To minimize kinetic energy, the matrix parameter to the control systemis set to be the manipulator mass matrix, and the scalar parameter isset to zero. That is,

W=M(q),  (8)

α=0.  (9)

In this case, when α=0, the vector parameter is not relevant. There aretwo classes for implementing this control strategy. For use with thebasic core, EcControlExpressionMassMatrix returns the manipulator massmatrix—M(q) in equation (8)—directly for use as weighting parameter W.The manipulator mass matrix is calculated using the method of CompositeRigid-Body Inertia (developed in “Efficient Dynamic Computer Simulationof Robotic Mechanisms,” M. W. Walker and D. E. Orin, Journal of DynamicSystems, Measurement, and Control, vol. 104, September 1982, pp. 205-211and named in Robot Dynamics Algorithms, R. Featherstone, Kluwer AcademicPublishers, 1987, both of which are hereby incorporated by reference),which is extended to apply to bifurcating arms. The same algorithm isused to calculate the mass matrix for minimum-kinetic-energy control asis used to calculate the composite-rigid-body dynamics. For use with theAB core, an A parameter corresponding to the manipulator mass matrix asW is calculated by treating the Jacobian-null-space basis vectors asaccelerations and calculating the corresponding joint torques. In thisexemplary embodiment, for this calculation the iterative Newton Euleralgorithm is used; for more information see “On-Line ComputationalScheme for Mechanical Manipulators,” J. Y. S. Luh, M. W. Walker, and R.P. C. Paul, Journal of Dynamic Systems, Measurement, and Control, vol.102, pp. 69-76, June 1980, which is hereby incorporated by reference.

Minimum Potential Energy Control

To minimize the potential energy of the arm, the gradient of thepotential energy is used. For articulated mechanisms, the gradient ofpotential energy is composed of gravitational joint torques. Tocalculate the gravitational joint torques (forces for prismatic joints)the composite rigid-body inertia of the links is used. This method isimplemented for the basic core in the classEcControlExpressionPotentialEnergyGradient.

Accuracy Optimization

For accuracy optimization in the presence of position errors in a singlejoint, an F value is provided for the basic core through the classEcControlExpressionError Sensitivity. For this calculation, a weightw_(i) is assigned to each of E end-effectors on the manipulator, and afocus joint, j, is identified. Then an error sensitivity function isdefined as follows:

$\begin{matrix}{{{f(q)} = {\sum\limits_{i = 1}^{E}{w_{i}{_{j}v_{i}}^{2}}}},} & (10)\end{matrix}$

where _(j)v_(i) is the velocity of end-effector i due to unit motion ofjoint j.

The gradient of this function is then found using finite differencing,and this is used as the vector parameter for the core velocity controlsystem. With the matrix parameter being a diagonal matrix and the scalarparameter a positive value, this serves to minimize the sensitivity toerrors.

Joint-Limit Avoidance

For joint-limit avoidance, for each joint, a rate term is independentlycalculated that is a polynomial function of the proximity to a limit.This is illustrated in FIG. 2, which shows how the vector elements forjoint-limit avoidance are set using polynomials with a user-configurableexponent and dead-zone. Class EcControlExpressionJointLimitAvoidanceimplements this weighting function. This is used with a negative scalarvalue to drive the manipulator away from joint limits.

Strength Optimization

Strength optimization is achieved by employing a technique based on thestrength formulation in “The Minimum Form of Strength in Serial,Parallel and Bifurcating Manipulators,” R. O. Ambrose and M. A. Diftler,Proceeding of the 1998 IEEE International Conference on Robotics andAutomation, Leuven, Belgium, May 1998, which is hereby incorporated byreference. In this discussion, torque is used in the general sense,meaning force for a sliding joint. For an arbitrary set of end-effectorforces, a vector of joint torques can be found. The goal of strengthoptimization is to adjust the manipulator joint positions in such a wayas to minimize the normalized joint torques resulting from the imposedforces while simultaneously placing the hand(s). This is accomplished bydirecting the joints toward the direction that yields the maximumreactive strength in the direction opposing the applied forces. Strengthhere is defined as the maximum force or moment that the manipulator canexert on its environment at the point of resolution of an end effector.

The basic formulation of velocity control used to relate joint rates tohand velocity is given by equation (1). Again, V is a concatenation ofhand velocities, {dot over (q)} is the vector of joint rates and J(q) isthe manipulator Jacobian.

If friction and gravitational loads are neglected, the power flow at thejoints must equal power flow at the end effector. By relating handvelocities V and corresponding wrench loads F with joint rates {dot over(q)} and joint torques T, this property is expressed as follows:

v^(T)F=−{dot over (q)}^(T)T  (11)

Using equation (1) with the fact that equation (11) must be true for allvalues of {dot over (q)} yields

T=−J(q)^(T)F  (12)

Equation (12) relates wrench loads exerted on the system with jointtorques required to prevent manipulator motion due to those loads.

When assessing the total strength of a manipulator, torque capacitiesfor each joint must be known. The torque capacity is typically limitedby the motor's ability to supply torque or the gearhead's ability totransmit torque to the joint. The torque capacity {circumflex over (T)}for a given joint can be expressed as

{circumflex over (T)}=min(t _(a) G,t _(t))  (13)

where t_(a) is the actuator (corresponding to motor) torque, G is thegear ratio (the ratio of actuator velocity to joint velocity—typically Gis greater than unity) and t_(t) is the transmission torque. The torquecapacity for a joint may differ depending on direction.To optimize manipulator configuration for strength, let x be a desiredend-effector force, as would be used with F=x in equation (12). Let{circumflex over (T)}_(max) _(j) be the maximum torque capacity, and{circumflex over (T)}_(min) _(j) be the minimum torque capacity (thelargest magnitude in the negative direction) for joint j. For an n-jointsystem, a measure of strength f(q) is given by

$\begin{matrix}{{{f(q)} = {\sum\limits_{j = 0}^{n - 1}\left\lbrack \frac{{T_{j} - \mu_{j}}}{{\overset{\Cap}{T}}_{j}} \right\rbrack^{r}}},} & (14)\end{matrix}$

where T_(j) is the actual torque required to achieve F=x using eq. (12),μ_(j) is the mean of the torque capacity of joint j, given by:

$\begin{matrix}{{\mu_{j} = \frac{{\overset{\Cap}{T}}_{\max_{j}} + {\overset{\Cap}{T}}_{\min_{j}}}{2}},} & (15)\end{matrix}$

and ∥{circumflex over (T)}_(j)∥ is the range of torque capacity forjoint j, given by:

∥{circumflex over (T)} _(j) ∥={circumflex over (T)} _(max) _(j)−{circumflex over (T)} _(min) _(j)   (16)

The user-defined parameter r specifies the order of the function tooptimize. Note that r=2 corresponds to a sum of squares approach and r→∞approaches an infinity-norm approach, where the strength is defined onlyby the weakest joint.

The gradient of ƒ(q) is found using finite differencing, forcing thejoints in the direction of minimum normalized torque and, hence, maximumstrength. This gradient is calculated in the expression tree using theclass EcControlExpressionStrengthOptimization. Because it is a gradient,it is for direct use with the basic core velocity control system. It canbe converted to a B parameter for use with an AB core by combining itwith an EcControlExpressionVectorToAB expression.

Statistical Error Reduction

For accuracy optimization in the presence of high frequency noise onmultiple joints, an approach is used that exploits the statisticalproperties of the joint noise as well as the Euclidean Space nature ofthe tools the manipulator is using.

The Function for Optimization

Let the velocity error in the manipulator's joints, {dot over (q)}_(e),be a random variable having covariance C_(e). That is, the expectedvalue of the quadratic form is given by

E[{umlaut over (q)} _(e) ·{dot over (q)} _(e) ^(T) ]=C _(e)  (17)

It is assumed that the controller prevents low-frequency drift toenforce E({dot over (q)}_(e))=0. It is desired to minimize the expectedvalue of a quadratic measure of the hand velocities due to this error.Let the error in the hand velocities be V_(e), now also a randomvariable. From equation (1), the relationship between {dot over (q)}_(e)and V_(e) is

V _(e) =J(q){dot over (q)} _(e)  (18)

Let a measure of the hand error be defined as

μ_(e)=V_(e) ^(T)AV_(e)  (19)

for some prescribed constant matrix A. Note μ_(e) is a random variable.The goal, then, is to minimize the expected value of μ_(e). That is, thefunction to be minimized is given by

ƒ(q)=E[μ _(e)]  (20)

Combining the three above equations gives

ƒ(q)=E[{dot over (q)} _(e) ^(T) J ^(T) AR{dot over (q)} _(e)]  (21)

To put this in a form that can be used for optimization, the followingformula is used, which applies for any column vector x and any matrix M,

x ^(T) MX=tr(M×x ^(T)).  (22)

This gives

ƒ(q)=E[tr(J ^(T) AJ{dot over (q)} _(e) {dot over (q)} _(e) ^(T))]  (23)

Here, tr(·) is the sum of the diagonal entries (the trace operator).Using E({dot over (q)}_(e))=0, this gives

ƒ(q)=tr(J ^(T) AJC _(e)),  (24)

a straightforward function of configuration.

Evaluating the A Matrix

The matrix A as used above can be set in a number of ways. By way ofexample, one approach is to use Euclidean-Space regions of interestrigidly attached to the hands. Such a region might be the physicalextent of a tool, for example.

Let the Euclidean-space region be labeled Ω. Then μ_(e) can be cast tominimize an integral of the velocity error squared over Ω. Let {rightarrow over (r)} be a point in Ω, and let p({right arrow over (r)}) be anonnegative interest density function defined over Ω (specifying, forexample, that the tip of a screwdriver is more important than theshaft). Let {right arrow over (v)}_(e)({right arrow over (r)}) bedefined as the error velocity at point {right arrow over (r)}, a randomvariable. These are illustrated in FIG. 3, which shows the terms used tomeasure the statistical error of a Euclidean Space region in the regionrigidly attached to one end-effector frame.

With this, what is desired is to minimize

ƒ(q)=E└∫ _(Ω)ρ({right arrow over (r)})∥{right arrow over (v)}_(e)({right arrow over (r)})∥² dω┘.  (25)

That is, the expected value of a weighted integral of the velocity errorsquared of all the points in the region of interest. This is acomprehensive and intuitive measure.

It is not practical, in many cases, to calculate the integral shown ineq. (25) in real time. However, there is a way to cast it into the formof eq. (19), which will allow eq. (24) to be used with no on-lineintegration. To do this, an analogy is used between eq. (25) and kineticenergy of a rigid body. If {right arrow over (v)}_(e)({right arrow over(r)}) were the velocity over the region Ω and ρ({right arrow over (r)})were the mass, then the integral in eq. (25) would provide twice thekinetic energy of the body.

Let the following be defined in analogy to mass, first moment ofinertia, and second moment of inertia:

m _(Ω)=∫_(Ω)ρ({right arrow over (r)})dω,  (26)

h _(Ω)=∫_(Ω)ρ({right arrow over (r)}){right arrow over (r)}dω,  (27)

I _(Ω)=∫_(Ω)ρ({right arrow over (r)})R ^(T) Rdω,  (28)

where R is the cross product matrix for {right arrow over (r)}(i.e.,Rx={right arrow over (r)}×x for any column vector x).

With these values, let the following matrix be defined:

$\begin{matrix}{{J_{\Omega} = \begin{bmatrix}{m_{\Omega}I} & H_{\Omega} \\H_{\Omega}^{T} & I_{\Omega}\end{bmatrix}},} & (29)\end{matrix}$

where I is the 3×3 identity matrix and H_(Ω) is the cross-product matrixfor h_(Ω). This produces, for a single frame end effector, thefollowing:

∫_(Ω)ρ({right arrow over (r)})∥{right arrow over (v)} _(e)({right arrowover (r)})∥² dω=V _(e) ^(T) J _(Ω) V _(e)  (30)

Equation (30) the same form as eq. (19), allowing the optimization ofeq. (24) to be performed using eq. (24) with A=J_(Ω).

In general, there will be N end effectors, and the matrix A will beblock diagonal:

$\begin{matrix}{A = {\begin{bmatrix}\left\lbrack J_{\Omega,1} \right\rbrack & \; & \; & 0 \\\; & \left\lbrack J_{\Omega,2} \right\rbrack & \; & \; \\\; & \; & \ddots & \; \\0 & \; & \; & \left\lbrack J_{\Omega,N} \right\rbrack\end{bmatrix}.}} & (31)\end{matrix}$

Implementation of Statistical Error Reduction

Statistical error reduction is implemented through theEcControlExpressionError Reduction class. This gives a vector weight foruse with the basic core velocity control system. It can be converted toa B parameter for use with an AB core by combining it with anEcControlExpressionVectorToAB expression.

End-Effector Descriptions

The array V, as used above, represents the motion of all themanipulator's end effectors. A special class can hold the description ofan end-effector set, which contains any number of end effectors. The endeffectors can represent any type of constraint. Implemented end-effectortypes include frame, 3D point, 3D orientation, 2D point, center of mass,and spatial momentup. More types can be added using the toolkit or theplugin interface.

To allow this general approach, many of calculations needed for velocitycontrol are performed in the end-effector class. The public methods thatmust be implemented to define a new end effector are given in the tablebelow.

TABLE Member functions that are implemented to define a new type of endeffector. Member Function Meaning doc Returns the end effector's degreesof constraint. For a point end effector, it returns 3. For a frame endeffector, 6. insertJacobianComponent Builds a strip of the Jacobian. Theheight of the strip equals the value returned by doc.insertSparsityComponent Builds a strip of the sparsity description ofthe Jacobian. A value of true in this strip means the correspondingposition in the Jacobian is always zero. The height of the strip equalsthe value returned by doc. calculatePlacement Calculates a placementvalue for the end effector. The placement is described through anEcCoordinateSystemTransformation, which may have different meanings fordifferent end-effector types. calculateVelocity Calculates end-effectorvelocity. The result is a real vector of length equal to the valuereturned by doc. The velocity will have different meanings for differentend-effector types. calculateAcceleration Calculates end-effectoracceleration. The result is a real vector of length equal to the valuereturned by doc. The acceleration will have different meanings fordifferent end- effector types. filterEndEffectorVelocity Calculates anend-effector velocity that drives the end effector toward a guideplacement. The guide frame is always represented in system coordinates.minimumTime Calculates the minimum time that will be used to move fromone frame to another. difference Calculates a measure of the differencebetween two placement frames that uses Euclidean distance as itsbaseline. That is, the difference between two frames is the Euclideandistance between them plus optional additional factors related toorientation change.

Through this approach, any new end effector can be added as a subclassof EcEndEffector, provided these member functions are implemented in thenew class. These member functions allow the end-effectors to createtheir own Jacobians and position controllers. Subclasses ofEcEndEffector include EcFrameEndEffector, EcPointEndEffector, andEcXyEndEffector.

Any of the foregoing control strategies, and others, can be implementedusing a dynamically updatable control system of the type described inU.S. Pat. No. 6,757,587 of English et al. (“Method and Apparatus ForDynamically Reprogramming Remote Autonomous Agents”), which is herebyincorporated by reference in its entirety. For example, mission controlcan select expressions or values for one of more of α, W, and ƒcomponents of the control system equations, expressing those componentsas an XML file and communicating it to a control system (e.g., a robot).The robot can validate the XML file using an on-robot copy of theschema, and use the validated XML to create in memory, at run time andusing a recursion and a creator function, a code tree structurerepresentation of the components of which the XML file provides anexpression or a value.

C. DERIVATION OF CONTROL MODEL

The exemplary control model disclosed above, including coordinatingcontrol of mobile or floating bases concurrently with an arm, will nowbe described in more detail.

The use of mobile manipulators can be modeled using a reference-framehierarchy. In the framework, the location of the base of eachmanipulator can be specified through a sequence of transformations, asillustrated in FIG. 4. The entire manipulator system is represented inthe system frame, which is defined with respect to a universal referenceframe as part of the environmental specification. Then, each manipulatorhas a static base frame whose location with respect to the system frameis specified as part of the manipulator description. The (dynamic)location of the manipulator with respect to the static base frame isspecified as part of the state, through the dynamic base frame. FIG. 4illustrates the location of a manipulator's base frame. The dynamic baseframe location is identified through a sequence of transformations. Formobile manipulators, the motion of the dynamic base frame is integratedwith the motion of the joints.

The dynamic base frame specifies the location of the base link on themanipulator, as illustrated in FIG. 5. FIG. 5 illustrates how one linkon the manipulator serves a special role as the base link. It does nothave a joint associated with it. Instead, the location of the dynamicbase frame determines its location. In an exemplary softwareimplementation, in the code the base link is a subclass of the normallink type that stores information on the entire manipulator.

Aspect Link

In addition to the base link, there is another link that can be used asa reference for controlling the manipulator when the base is not fixed.This is the aspect link, as shown in FIG. 6. As opposed to the baselink, the aspect link is conceptual (i.e., any link can be assigned asthe aspect link). As an object in code, it is just another link. FIG. 6illustrates how one link on the manipulator serves a special conceptualrole as the aspect link. It is a regular link, distinguished as theaspect link only through the control system. For a free-floating base,the frame velocity V_(a) is used for calculating kinematic control.

Moving-Aspect Jacobian

Each end effector is defined as rigidly attached to some link. Thisend-effector link can be any link of the manipulator, and is typicallyseparate from both the base link and the aspect link. This is shown inFIG. 7, which depicts how the end-effector link, can be rigidly attachedto the end effector for which kinematic control is desired.

For end-effector position {right arrow over (r)}_(e) and base position{right arrow over (r)}_(b), let

{right arrow over (p)} _(b→e) ={right arrow over (r)} _(e) −{right arrowover (r)} _(b)  (32)

And, for any vector {right arrow over (p)}, let P be defined as:

$\begin{matrix}{{P = \begin{bmatrix}I & {{- \overset{\rightarrow}{p}}x} \\0 & I\end{bmatrix}},{with}} & (33) \\{P^{- 1} = {\begin{bmatrix}I & {\overset{\rightarrow}{p}x} \\0 & I\end{bmatrix}.}} & (34)\end{matrix}$

With this definition, the frame velocity of the end effector is givenby:

V _(e) =J _(e) {dot over (q)}+P _(b→e) V _(b)  (35)

where J_(e) is the manipulator Jacobian. All quantities here are assumedto be represented in the same frame.

Similarly, for the aspect link the frame velocity of the aspect frame isgenerally:

V _(a) =J _(a) {dot over (q)}+P _(b→a) V _(b)  (36)

where J_(a) is the manipulator Jacobian for the aspect frame. The matrixP_(b→a) can never be singular, allowing:

V _(b) =P _(b→a) ⁻¹(V _(a) −J _(a) {dot over (q)}).  (37)

Combining (45) and (47) gives

V _(e) =J _(e) {dot over (q)}+P _(b→e) P _(b→a) ⁻¹(V _(a) −J_(a{dot over (q)})),  (38)

Using P_(b→e)P_(b→a) ⁻¹=P_(a→e) and collecting terms gives

V _(e)=(J _(e) −P _(a→e) J _(a)){dot over (q)}+P _(a→e) V _(a)  (39)

This allows the moving-aspect Jacobian to be defined as

J* _(e) ^(a) =[J _(e) −P _(a→e) J _(a) |P _(a→e)].  (40)

Let {dot over (q)}*_(a) be defined by concatenating {dot over (q)} andV_(a) as follows:

$\begin{matrix}{{{\overset{.}{q}}_{2}^{*} = \begin{bmatrix}\overset{.}{q} \\\ldots \\V_{a}\end{bmatrix}}{Then}} & (41) \\{V_{e} = {J_{e}^{*a}{\overset{.}{q}}_{2}^{*}}} & (42)\end{matrix}$

gives the manipulator Jacobian equations when the aspect link is moving.

Using this with

$\begin{matrix}{{{\overset{.}{q}}_{2}^{*} = {\left\lbrack \frac{J_{e}^{*a}}{N_{J_{e}^{*a}}^{T}W_{a}^{*}} \right\rbrack^{- 1}\left\lbrack \frac{V_{e}}{{- \alpha}\; N_{J_{e}^{*a}}^{T}F_{a}^{*}} \right\rbrack}},} & (43)\end{matrix}$

where, for N links, W*_(a) is an (N+6)×(N+6) symmetric matrix and F* isan (N+6)×1 column vector, gives a value of {dot over (q)}* that producesthe desired V_(e) while minimizing

μ=½({dot over (q)}* _(a))^(T) W* _(a) {dot over (q)}* _(a)+({dot over(q)}* _(a))^(T) F* _(a)  (44)

This is a desirable result. An arbitrary quadratic function of q and V,can be minimized this way. However, to calculate equation (43) directlyinvolves calculating a special Jacobian as a function of the attributeframe, which can be disadvantageous in some cases.

To avoid this, there is an alternative representation that allows theregular manipulator Jacobian to be used, yet achieve the same result.More specifically, from equation (36) the following equation holds:

V _(b) =P _(a→b) V _(a) −P _(a→b J) _(a) {dot over (q)},  (45)

Let T_(a-pb) be defined as

$\begin{matrix}{{T_{a\rightarrow b} = \begin{bmatrix}I & 0 \\{{- P_{a\rightarrow b}}J_{a}} & P_{a\rightarrow b}\end{bmatrix}},{and}} & (46) \\{{\overset{.}{q}}_{b}^{*} = {\begin{bmatrix}\overset{.}{q} \\\ldots \\V_{b}\end{bmatrix}.}} & (47)\end{matrix}$

This gives

{dot over (q)}* _(b) =T _(a→b) {dot over (q)}* _(a)  (48)

Let J*_(e) be defined as

J* _(e) =[J _(e) |P _(b→e)].  (49)

This can then be used with

$\begin{matrix}{{\overset{.}{q}}_{b}^{*} = {\left\lbrack \frac{J_{e}^{*}}{\; {{N_{J_{e}^{*}}^{T}\left( T_{a\rightarrow b}^{- 1} \right)}^{T}W_{a}^{*}T_{a\rightarrow b}^{- 1}}} \right\rbrack^{- 1}\left\lbrack \frac{V_{e}}{{- \alpha}\; {N_{J_{e}^{*}}^{T}\left( T_{a\rightarrow b}^{- 1} \right)}^{T}F_{a}^{*}} \right\rbrack}} & (50)\end{matrix}$

to give the same result as (43).

Let

W*=(T _(a→b) ⁻¹)^(T) W* _(a) T _(a→b) ⁻¹  (51)

and

F*=(T _(a→b) ⁻¹)^(T) F* _(a)  (52)

Note T_(a→b) ⁻¹ has the following convenient form:

T_(a→b) ⁻¹=[J_(a) ^(I)P_(b→a) ⁰].  (53)

With this,

$\begin{matrix}{{\overset{.}{q}}_{b}^{*} = {\left\lbrack \frac{J_{e}^{*}}{\; {N_{J_{e}^{*}}^{T}W^{*}}} \right\rbrack^{- 1}\left\lbrack \frac{V_{e}}{{- \alpha}\; N_{J_{e}^{*}}^{T}F^{*}} \right\rbrack}} & (54)\end{matrix}$

gives the same manipulator motion as (43), but without the need tocalculate J*_(e) ^(a). The choice of aspect frame is incorporated onlyinto the parameterization values W* and F* (note the structure of (54)is the same as that of (43)).

Relative Jacobians

An end effector can be defined relative to another frame on the samemanipulator. In this case, the Jacobian equation is the same whether ornot the aspect link is mobile. If a relative end effector is defined, anew special link is added to the manipulator, as shown in FIG. 8. FIG. 8illustrates how an end effector can be defined relative to any otherlink.

Let ^(S)V_(e) be the frame velocity of the end-effector with respect toand expressed in the system frame, with linear velocity ^(S)V_(e) andangular velocity ^(S)ω_(e). The point of application for these values is{right arrow over (r)}_(e). Similarly, Let ^(S)V_(r) be the framevelocity of the reference frame with respect to and expressed in thesystem frame, with linear velocity ^(S)v_(r) and angular velocity^(S)ω_(r). The point of application for these values is {right arrowover (r)}_(r).

With these definitions, Jacobians J_(e) and J_(r) exist such that

^(S) V _(e) =J _(e) {dot over (q)}+P _(b→e) ^(S) V _(b)  (55)

and

^(S) V _(r) =J _(r) {dot over (q)}+P _(b→r) ^(S) V _(b)  (56)

Then the velocity of the end effector with respect to the relative frameis

^(r) V _(e)=^(S) V _(e) −P _(e→r) ^(S S) V _(r)  (57)

The effect of V_(b) cancels, leaving the following Jacobian:

J ₁ =J _(e) −P _(e→r) ^(S) J _(r)  (58)

This has no dependence on the motion of the base.

Specification of Mobile-Base Parameters

The description of the base, aspect, relative, and end-effector linksshould be given in the control-system description. Whether or not theaspect link is moving or stationary should be specified. Possiblemethods for specifying these are through the manipulator system, themanipulator state, the velocity control system, or the end-effectors.

In some implementations, the mobile-base parameters are specified asfollows:

TABLE Method by which the various components are specified. VelocitySystem Control End Effector Base Link Jacobian End Effector Link BaseMobility Aspect Link Relative Link Aspect Mobility

D. EXEMPLARY HARDWARE/SOFTWARE IMPLEMENTATIONS

FIG. 9 depicts an exemplary block diagram of a control system 900 inwhich input signals 912 come from, e.g., user input device 902. A userinput device can include, e.g., any input device known in the artassociated with a computer, such as a keyboard or mouse, screens (e.g.,touchscreens), joysticks, steering wheels, and other input devices whichmay be tailored to the apparatus they are controlling. The input signals912 generated by the user input device 902 can be used to input and/orindicate the desired control strategy for the control system (e.g., asrepresented by parameters α, W, and ƒ, described above) and/or thedesired motion (e.g., velocity vectors) of the apparatus (e.g., bymanipulation of a joystick, for example). Input signals 912 can also begenerated by sensors 904, such as machine vision, contact sensors,position sensors (e.g., GPS), gyroscopes, and so on.

In the exemplary embodiment of FIG. 9, input signals 912 are fed intodigital data processor(s) 906, which can be embodied in any general orspecial purpose computer. In many cases the user input device 902 anduser display 918 can be integrated with the digital data processor 906in a manner known in the art to create a familiar personal computer,laptop, PDA, and so on. The digital data processor 906 can take the formof an operator control unit (OCU) with a user interface (e.g., asoftware interface providing a GUI or text-based interface) inaccordance with the teachings herein. In some embodiments, the digitaldata processor 906 can be communicatively coupled, e.g., via a network,wirelessly, or otherwise, to the apparatus (e.g., apparatus 920) whichit controls (e.g., for communicating control signals 914). In otherembodiments, the digital data processor 906 can be embedded, e.g.,within the controlled apparatus.

The digital data processor 906 includes algorithms stored on acomputer-readable medium, e.g., magnetic, optical, solid-state or otherstorage medium. Such algorithms can effect control in accordance withthe control principles disclosed herein and may include standardoff-the-shelf software packages for implementing certain functionality.Such software packages may include, for example, the Actin (foreffecting control) and Selectin (for machine vision) software toolkitsdeveloped by Energid Technologies of Cambridge, Mass., USA. One or moreprocessors of any type known in the art can execute the instructionsembodied in algorithmic form in view of the input signals received fromthe user input devices and/or sensor devices to effect control of theapparatus.

Control signals 914 generated by the digital data processor can becommunicated to the controlled apparatus 920 and more particularly, inmany cases, to one or more actuators 916 in such apparatus. Thecontrolled apparatus 920 can have its own local intelligence, e.g.,processing capability, for interpreting the control signals 914,translating the control signals 914 to a form appropriate for theactuators 916, and/or, in some cases, determining some or all of thecontrol signals 914 itself. Actuators 916 may take the form of anydevice capable of being controlled or instructed, including motors,servos, pneumatic devices, magnetic devices, electromagnets, and so on.In some cases, the control signals 914 may be transformed to theappropriately (e.g., to an appropriate voltage, current, orcommunications protocol) for the actuators 914 to which they areassigned. Regardless, the actuators operate under the direction of thecontrol signals 914 to effect the desired position changes and/or motionof each movable element of the controlled apparatus 920.

FIG. 10 depicts another block diagram of an exemplary top-level softwarearchitecture 1000 for controlling an apparatus such as a roboticapparatus and references exemplary software packages, such as theabove-referenced Actin and Selectin software packages. As shown in FIG.10, information is exchanged A) between the operator control unit (OCU)and the coordinator on the robotic apparatus, B) between the coordinatorand the robot Selectin-based vision module, C) between the coordinatorand the Actin-based force and position control, D) between thecoordinator and the grasping algorithm database, E) camera imagery fromthe robot platform, F) control interface for the robot platform (in somecases, a software interface provided by a robot manufacturer, e.g.,Aware 2.0 for iRobot's Packbot), and G) between the OCU and theSelectin-based vision module. The robot and interface, marked with a boxon the left, can be optionally real or simulated.

FIG. 11 depicts an exemplary flow-chart of steps that can be executed ina digital data processing system, e.g., as shown in FIGS. 9-10, thatfunctions in accordance with the teachings herein. As shown in FIG. 11,in step 1100 the system can receive data indicating the selected controlstrategy, e.g., as represented by a merit function to be optimized withparameters α, W, and ƒ as described above in connection with Equation(3). Further, the system can receive input signals indicating a desiredmotion at step 1102. Such input signals may take the form of desiredvelocity vector V for an end effector (e.g., as described above inconnection with equations (1) and (2)), which can take the form of anm-length vector. In step 1104, the system can generate control signals(in some cases the control signals can represent control vectorsrepresenting a temporal rate of position changes for each joint, asdescribed above in connection with {dot over (q)}, for example). Thecontrol signals can be optimized in accordance with the merit function(such as equation (3), for example) and/or selected control strategy. Inmany embodiments, the control signals can be generated using equation(2). In step 1106, the control signals can be applied to a roboticapparatus being controlled, e.g., to the software interfaces providedfor that apparatus, which can apply the proper signals to actuators toeffect movement (e.g., the position changes mentioned above.) In otherembodiments, the apparatus software interfaces may be omitted orunnecessary and the control signals applied to the actuators directly.At 1112, the apparatus exhibits the desired and optimized motion. Inmany cases, such motion involves simultaneously moving a mobile base andother movable elements (e.g., making up a manipulator).

E. EXAMPLES & SIMULATIONS

By way of further illustration, the following examples are included tofurther elucidate, among other things, the principles described above.The examples include exemplary simulations provided via softwaretoolkits developed in accordance with the teachings disclosed herein. Itshould be understood that the information presented in connection withthese examples is provided for illustrative purposes and is not intendedto necessarily show optimal results that can be achieved or that need tobe achieved by employing control systems and methods in accordance withthe teachings of this application.

Examples 1 and 2 Coordination Control For Mobile Base

Presented below are two simplified kinematic simulations—a onedegree-of-freedom sliding end effector and a planar end effector. Theterm end effector used here refers to controlling the base as an endeffector. In a kinematic simulation, any link, including the base, canbe treated as an end effector. Many different kinds of end effectors areimplemented in the Actin software toolkits. The naming of these endeffectors usually reflects kinematic restrictions rather than functions.For example, a point end effector indicates that the three dimensionalposition of the end effector needs to match that position with thedesired one (the origin of the guide frame in the GUI. The guide frameis a visualization of a set of three orthogonal axes that shows thedesired position and orientation for the end effector; for a point endeffector, the orientation does not matter. The frame end effector canmatch the guide frame in both position and orientation. In the GUI, theuser manipulates the guide frame via mouse or other input devices andthe selected end effector follows the guide frame in a different waybased on the different end effector type. Note that there can be morethan one end effector active at the same time. For example, if the baseof the robot has been assigned to a frame end effector and its gripperhas been assigned to another; then manipulating the gripper's guideframe will move the gripper around while keeping the base fixed.

This example illustrates coordinated motion between an arm and the base.In this case, the sliding end effector mentioned above will be assignedto the base and it will allow the base to move back and forth, noturning or side moves. The guide frame for the base's sliding endeffector is not moving; we only move the gripper's guide frame. Theplanar end effector works in a similar fashion. Instead of allowing thebase moving back and forth; the base can move freely on the x-y plane(with z pointing up). Note that the tracks will not be able to move sideways, so any velocity in the side direction needs to be filtered out.The following sub-sections explain in detail how these end effectors andthe filter are implemented.

Example 1 Implementation of Sliding End Effector for the Base

In this example the mobile base is modeled after an end effector whichexhibits sliding motion. In other words, constraints are placed on themovement of a mobile base, e.g., to model those constraints that mightexist with real bases.

The relation between the end effector velocity and the joint velocitycan usually be written as the Jacobian equation:

V=J(q){dot over (q)},  (59)

where V is an m-length vector representation of the motion of the endeffectors; q is an n-length vector of joint and base positions (with qbeing its time derivative); and J is the m×n Jacobian matrix as functionof q. This example will illustrate control of the mobile base as asingle degree of freedom mechanism that can move forward and backward.One way to achieve this is to model the mobile base as a six degree offreedom object and apply a five degree of freedom constraint on it. Wedefine a frame attached to the base and assume the base can only move inthe frame's x direction; its orientation and y, z coordinates will notchange. This part of Jacobian is written as

$\begin{matrix}{{\begin{bmatrix}v_{e} \\w_{e}\end{bmatrix} = {{\overset{\Cap}{J}(q)}\begin{bmatrix}\theta_{0} \\\vdots \\\theta_{k} \\v_{b} \\w_{b}\end{bmatrix}}},} & (60)\end{matrix}$

where y_(e) and w_(e) are the linear and angular velocity of the endeffector frame in the system coordinates; v_(b) and w_(b) are the linearand angular velocity of the base in the system coordinates. Note thatĴ(q) is only part of the whole Jacobian matrix that corresponding to theconstraint of this end effector. The relation between v_(e), w_(e) andV_(b) w_(b) can be written as

$\begin{matrix}{\begin{bmatrix}v_{e} \\w_{e}\end{bmatrix} = {\begin{bmatrix}{v_{b} + {w_{b} \times^{b}p_{e}}} \\w_{b}\end{bmatrix} = {\begin{bmatrix}I & \begin{bmatrix}0 & p_{z} & {- p_{y}} \\{- p_{z}} & 0 & p_{x} \\p_{y} & {- p_{x}} & 0\end{bmatrix} \\0 & I\end{bmatrix}\begin{bmatrix}v_{b} \\w_{b}\end{bmatrix}}}} & (61)\end{matrix}$

where p_(e)=[p_(x) p_(x) p_(x)]^(T) is the linear transformation fromthe base frame to the end effector frame represented in the base frame.Since the constraints are given in the end effector frame and the endeffector velocities are calculated in the system coordinates, thesevectors will be converted into the end effector's frame, as follows:

$\begin{matrix}{\begin{bmatrix}v_{f} \\w_{f}\end{bmatrix} = {\begin{bmatrix}{Rv}_{e} \\{Rw}_{e}\end{bmatrix} = {\begin{bmatrix}R & 0 \\0 & R\end{bmatrix}{{\overset{\Cap}{J}(q)}\begin{bmatrix}\theta_{0} \\\vdots \\\theta_{k} \\v_{b} \\w_{b}\end{bmatrix}}}}} & (62)\end{matrix}$

Therefore the portion of the resulting Jacobian matrix for this endeffector is obtained by taking the last five rows of the matrix

$\begin{bmatrix}R & 0 \\0 & R\end{bmatrix}{{\overset{\Cap}{J}(q)}.}$

This new partial Jacobian matrix is of size 5×(k+6).

The implementation of this class is in Actin-controlEcSlidingEndEffector. The robot (e.g., iRobot's Packbot) base endeffector frame has x pointing forward, z pointing up and the y pointingthe side way. With this end effector defined to the base, the base canmove forward and backward; which add one more degree of freedom to thesystem and makes the arm a 6 degree of freedom arm instead of 5.

Example 2 Planar Motion Base Simulation

This example for the mobile base kinematic simulation illustrates planarmotion; where the platform (mobile base) can move in the x-y plane androtate about the z-axis. An exemplary two-wheeler model for this isshown in FIG. 12. FIG. 12 illustrates the model for mobile platform on a2-dimensional plan. Distance between the wheels is equal to w, wheelradius is equal to r.

With reference to FIG. 12, while the frame attached to the axel hasthree degrees of freedom (x, y and θ_(z)), the joints only have twodegrees of freedom (θ₀ and θ₁). The relation between the joint velocityand the frame velocity can be written as:

$\begin{matrix}{\begin{bmatrix}\overset{.}{x} \\\overset{.}{y} \\{\overset{.}{\theta}}_{z}\end{bmatrix} = {\begin{bmatrix}\frac{r\; \cos \; \theta_{z}}{2} & \frac{r\; \cos \; \theta_{z}}{2} \\\frac{r\; \sin \; \theta_{z}}{2} & \frac{r\; \sin \; \theta_{z}}{2} \\{- \frac{r}{w}} & \frac{r}{w}\end{bmatrix}\begin{bmatrix}{\overset{.}{\theta}}_{0} \\{\overset{.}{\theta}}_{1}\end{bmatrix}}} & (63)\end{matrix}$

Note that the Jacobian matrix relating the linear velocity {dot over(x)}, {dot over (y)} of the end effector and the joint angle velocitiesθ₀ and θ₁ is singular; meaning position error can not be correctedthrough end effector control if there is no angular error.

One way to solve this problem is to correct the position error byrotating the platform to the direction of desired linear velocity, thenmove the platform forward or backward. Once the position error isminimized, the angular error can be corrected. This procedure can beimplemented in our control software using an end effector velocityfilter. Instead of using equation (63), we can follow the similar stepsas in the previous example. First, we form the Jacobian matrix for theend effector velocity represented in the end effector frame as inequation (62). That is, we treat the base as a 6 degree of freedomobject in space. Then we let the base move freely in x, y and θ_(z) bydropping the first ({dot over (x)}), second ({dot over (y)}) and thesixth (θ_(z)) rows of the Jacobian matrix

${\begin{bmatrix}R & 0 \\0 & R\end{bmatrix}{\overset{\Cap}{J}(q)}};$

which results in a Jacobian matrix of size 3×(k+6). The implementationof this class is in actin-control EcPlanarEndEffector

In this example, a constraint is placed on the base such that the trackscannot move in y (side ways). To achieve this, the planar end effectorcan be combined with a velocity filter to restrict the mobile platformfrom sideways (y) motions. The resulting velocity of this filter isachievable with tracks (e.g., a tracked mobile robot).

The planar end effector acts as a restriction which adds 6 elements tothe joint positions q in the Jacobian equation:

V=J(q){dot over (q)}  (64)

and 3 velocity terms to V. For this portion of the Jacobian equation, itcan be written as

$\begin{matrix}{{\begin{bmatrix}z \\w_{x} \\w_{y}\end{bmatrix} = {{\overset{\Cap}{J}(q)}\begin{bmatrix}\theta_{0} \\\vdots \\\theta_{k} \\v_{b} \\w_{b}\end{bmatrix}}},} & (65)\end{matrix}$

where z is the linear velocity represented in the end effector frame (aframe attached to the mobile platform). w_(x) and w_(y) are the angularvelocities of the end effector frame in the end effector coordinates;v_(b) and w_(b) are the linear and angular velocities of the DH frame ofthe mobile platform in the system coordinates. Note that there is a DHframe and a primary frame defined in each link. DH frames are definedfor the kinematics and primary frames are the coordinates system definedfor the CAD model of each link. There can be offset between the DH frameand the primary frame in each link. Also note that the end effectorframe is defined with respect to the primary frame. Therefore whenfiltering the joint velocities for the base, v_(b) and w_(b) given inthe DH frame, we need to convert them to velocity of the end effectorframe represented in the end effector frame. Exemplary pseudo code islisted as below:

EEToDHInSystem = baseEEFrameInSystem.translation( ) −baseDHFrameInSystem.translation( ); EEVelocityInSystem =DHVelocityInSystem.transformBy(EEToDHInSystem); EEVelocityInEE =EEVelocityInSystem.transformBy(baseEEFrameInSystem.orientation( ).-inverse( )); filteredEEVelInEE = EEVelInEE; // only thing we did here issetting the side velocity y to 0 filteredEEVelInEE.setLinearY(0.0); //convert velocity back in system coordinates. filteredEEVelInSystem = filteredEEVelInEE.transformBy(baseEEFrameInSystem.orientation( )); //calculate the corresponding velocity at the DH framefilteredDHVelInSystem =    filteredEEVelInSystem.transformBy(−EEToDHInSystem);

In this example, the side movement velocity can be set to 0. Dependingon the desired behavior of the tracks, different behaviors can beintroduced as needed by adding new types of velocity filters. Specialattention can be paid to the different frames (primary, DH and endeffector frames). Denavit-Hartenberg (DH) parameters specify thereference frame of any link in a robotic system with respect to anadjacent link. In this way, for a connected chain of links, the positionand orientation of each link is specified relative to the position andorientation (reference frame) of an adjacent link.

One exemplary filter is a joint rate filter implemented in the class inActin-control EcControlExpressionJointRateFilter. The planar tracksfilter and the joint rate filter are applied sequentially. The endeffector and the end effector frame defined with respect to the primaryframe can be changed on the fly through the GUI. The end effector dialogallows the user to select a manipulator in the system and the desiredlink in the manipulator. In this case (see FIG. 13) we want to assign aplanar end effector to the base. A pull down menu, illustrated in FIG.13 shows a dialog box for assigning end effectors and setting upparameters, which has a variety of implemented end effectors. For eachtype of end effector there is a list of associated parameters. The usercan change the parameter values through the dialog box.

A demo is created with the newly implemented sliding end effector andthe planar end effector combined with the planar tracks filter todemonstrate the coordinated arm and mobile platform motion. The arm ormanipulator (e.g., an explosive ordinance disposal arm or EOD) istypically operated with the robot base fixed. The new control schemeprovides a way to operate them together. This can have the advantage ofgiving the operator a single interface that is more intuitive. Inaddition, the integrated system has more degrees of freedom thanoperating the base and arm separately. This higher degree of freedomprovides many benefits when dealing with collision avoidance, jointlimit avoidance and many other secondary goals. In the simulation weassigned a point end effector (restricts just the location but not theorientation) to the camera and a planar end effector to the base. Wemoved the desired camera end effector position by dragging a guide framein the viewer and the camera follows. In the simulation it can be seenthat:

-   -   1. The integrated system has more degrees of freedom, has a        larger workspace, and can be controlled through a single user        interface.    -   2. The planar end effector and the planar tracks filter        effectively constraint the mobile base to behave like a tracked        mobile platform; it can move back and forth and can spin, but        cannot move sideways.    -   3. We defined a rough bounding volume for each link for        self-collision avoidance. The arm will automatically avoid        hitting any other link in the system without the operator's        attention.

Operator Control Unit (OCU) With Multiple Views

In one embodiment, an operator control unit can provide a user withmultiple views, such as an on-the-arm camera view and a fish-eye viewfrom a mobile platform. A user can select from different modes (drive,manipulate . . . etc.) to finish the goal. The arm may need to be drivendifferently when the camera on the arm is looking in differentdirections, in some cases. Through experiences drawn from human computerinterfaces, often WYSIWYG (what-you-see-is-what you-get) can provide anintuitive way of connecting the human and the machine. It can beadvantageous to provide the operator a way to control the arm withrespect to an individual camera view (e.g., with respect to the localcoordinate frame of that individual camera view). For example, upon aparticular view being brought into focus (e.g., via view selection by auser), the user interface (e.g., the user interface software) can switchto the mode that is tied to that current view; a seamless mode change.The operator can give a desired end effector pose with respect to thecurrent camera view; the software can translate the command to targetjoint angles automatically without the operator's attention and/orinput. The operator can place the 3D cursor in each of the views exactlythe same way as in any other view.

For example, a viewer (e.g., Actin viewer) supports rendering multipleviews by defining multiple virtual cameras for simulation purposes.

For simulating or controlling a robot with mobile base, it isadvantageous in many cases to set up a camera view from the camera nearthe end of a robot arm or at an intermediate link along the arm. Forexample, we can set the arm camera to look at a gripper or other endeffector, but it can also be set to look at any other part of thesystem. A typical camera may pan and tilt with respect to an arm link.Alternate cameras are often placed on the mobile base to provide abroader view of the robot arms or of the environment around the robot.In addition, virtual cameras can be placed anywhere in the 3D simulationand control environment, allowing views of the simulated robot pose andcontrol of the actual robot from any view angle. In some embodiments,actual cameras can also be used, either in addition to or in place ofvirtual cameras.

The multiview approach can show multiple views of the robot on the OCUdisplay. The multiview OCU can allow for direct control of the robotfrom any of the views, using a single active guide frame among all viewsor by using separate guide frames per view. Direct control can be donein the local coordinate frame of the selected window, making itintuitive and flexible.

Extensible Framework for User Input with Multiple Shared Views

User input can be handled in a way that expands the ease of user inputfor all windows, including a main window as well as alternate windows,child windows, and/or sub windows. There is a new framework forregistering input handlers, which allows for expansion and modificationof the behavior of input events. A set of classes was created in theActin toolkit to provide several levels of input control. These classescan be extended or overridden to create custom behavior.

EcSGBaseInputHandler—provides base class for framework to set andretrieve camera position and orientation. This class does not provideuser input control; only explicit camera placement is allowed with thisclass. This corresponds with the previous behavior of EcSGWindow.

EcDefaultInputHandler—A general class that takes user input events, suchas from a mouse or keyboard or touchscreen and processes them for thecurrent window. Example default behavior is left mouse button controlscamera view (rotation and translation in x,y) and right mouse buttonmanipulates the end-effector; middle mouse controls zooming for cameraor end-effector (z translation). This is the default input handler forEcBaseViewerMainWidget.

Examples 3 & 4 Free Base Kinematics Simulations

These simulations involve free-base kinematics focusing on achievablerobot configurations, independent of inertia involved in moving to adesired configuration. FIG. 14 shows an example 21-joint (27-degree offreedom, including base motion) free-floating mechanism placing twoframe end effectors and two point end effectors. The mechanism has 21joints and 4 end effectors and is being kinematically controlled. Thissimulated mechanism is built into code base for use in self-tests of thefree-base code. It illustrates the ability of the toolkit to support avariety of complex mechanisms.

FIGS. 15A-15D show the 57-link Robonaut model moving in free-base mode.FIGS. 15A-15D represent a sequence of images (from the same viewpoint)of the Robonaut model in free-base mode sequentially placing fingers ofopposing hands. In free-base mode, all links on the manipulator are freeto move. The only constraints are the end effectors. The toolkit can beused to support hand-over-hand movement of Robonaut when it is fielded.

Example 5 Dynamic Simulation

This example involves dynamic simulations which include modeling ofmotion-related properties such as inertia. It illustrates how twodynamic simulation methods for fixed bases, the Articulated Body InertiaAlgorithm and the Composite Rigid-Body Inertia Algorithm, can beextended to support simulating free-base manipulators. The Order(N)Articulated Body Inertia algorithm is best for very large manipulators,while the Order(N³) Composite Rigid-Body Inertia algorithm is best forsmaller manipulators.

Composite Rigid-Body Inertia Simulation Algorithm

For fixed-base manipulators, dynamic simulation is implemented using anadaptation of the composite rigid-body algorithm for bifurcatingmanipulators. For more information on such algorithms, see M. W. Walkerand D. E. Orin, “Efficient Dynamic Computer Simulation of RoboticMechanisms,” Journal of Dynamic Systems, Measurement, and Control, 104,205-211, 1982 and A. Fijany and A. K. Bejczy, “An Efficient Algorithmfor Computation of Manipulator Inertia Matrix,” Journal of RoboticSystems, 7(1), 57-80, 1990, both of which are hereby incorporated byreference.

This algorithm runs in Order(N³) time, for N links

The fixed-base composite rigid-body inertia algorithm uses the followingdynamics equation:

τ=M(q){umlaut over (q)}+G(q)+B,  (66)

where τ is the column vector of joint torques/forces, M(q) is themanipulator inertia matrix, q is the vector of joint position, C(q)represents the Coriolis forces, G(q) represents gravitational forces,and B represents the effect of external forces applied to the arm'slinks. This equation, as shown, is only valid for a manipulator with afixed base. When the base is free, it must be modified.

For any frames i and j that are rigidly connected, Let ^(j)P_(j→i) bethe cross-product matrix for ^(J)p_(i→j), the vector from the origin offrame i to the origin of frame j, expressed in frame j. And let^(j)R_(i) be the rotation matrix expressing frame i in frame j. Usingthis, let the matrices _(F)T_(i→j) and _(A)T_(i→j) be defined asfollows:

$\begin{matrix}{{\;_{\;}{{}_{}^{}{}_{\left. i\rightarrow j \right.}^{}} = \begin{bmatrix}{{\;^{\;}}^{j}R_{i}} & 0 \\{\;^{j}{P_{j\rightarrow i}}^{j}R_{i}} & {\;^{j}R_{i}}\end{bmatrix}},{and}} & (67) \\{\;_{A}T_{i\rightarrow j} = {\begin{bmatrix}{\;^{j}R_{i}} & {\;^{j}{P_{j\rightarrow i}}^{j}R_{i}} \\0 & {\;^{j}R_{i}}\end{bmatrix}.}} & (68)\end{matrix}$

These transformations produce the following equalities:

F_(j)=_(F)T_(i→j)F_(i)  (69)

and

A_(j)=_(A)T_(i→j)A_(i)  (70)

For any rigid body, let {right arrow over (ƒ)} be the vector forceapplied to the link, {right arrow over (n)} be the moment, {right arrowover (ω)} be the angular velocity, {right arrow over (v)} be the linearvelocity, {right arrow over (ƒ)}_(a) be an a priori external forceapplied to the body, {right arrow over (n)}_(a) be an a priori momentapplied to the body, m be the mass, H be the cross-product matrix forthe first moment of inertia, and J be the second moment of inertia.Then, the force/moment equations are given by the following:

{right arrow over (ƒ)}=H ^(T) {dot over ({right arrow over (ω)}+{rightarrow over (ω)}×H ^(T) {right arrow over (ω)}+m{dot over ({right arrowover (v)}−{right arrow over (ƒ)} _(e)  (71)

{right arrow over (n)}=J{dot over ({right arrow over (ω)}+{right arrowover (ω)}×J{right arrow over (ω)}+H{dot over ({right arrow over(v)}−{right arrow over (n)} _(e)  (72)

Let the 6×6 rigid-body inertia be defined as follows:

$\begin{matrix}{I^{C} = {\begin{bmatrix}{mI} & H^{T} \\H & J\end{bmatrix}.}} & (73)\end{matrix}$

And let a bias frame force be defined as

$\begin{matrix}{B = {\begin{bmatrix}{\overset{\rightarrow}{\omega} \times H^{T}\overset{\rightarrow}{\omega}} \\{\overset{\rightarrow}{\omega} \times J\overset{\rightarrow}{\omega}}\end{bmatrix} + {\begin{bmatrix}{- {\overset{\rightarrow}{f}}_{a}} \\{- {\overset{\rightarrow}{n}}_{a}}\end{bmatrix}.}}} & (74)\end{matrix}$

With this, the rigid-body dynamics can be represented as

F=I ^(C) A+B.  (75)

Effect of Base Acceleration on Joint Torque

When the base link is free to move, the force on and acceleration of thebase affects the manipulator dynamics. The affect of the baseacceleration on any joint can be found by fixing all other joints andfinding the torque required by the joint to sustain the acceleration onthe composite rigid body outboard from the joint. This is illustrated inFIG. 16, which shows how in this model the torque produced on joint idue to the acceleration of the base is the torque required to accelerateall the outboard links from the joint. This is an additive term, foundby assuming all other joints are stationary.

For the base joint, let the a priori external force and moment bedivided into two components. Let {right arrow over (ƒ)}_(e) an externalforce applied to the base, {right arrow over (n)}_(e) be an externalmoment applied to the body, {right arrow over (ƒ)}_(m) be the forceapplied by child links to the base, and {right arrow over (n)}_(m) bethe moment applied by child links. With this, {right arrow over(ƒ)}_(a)={right arrow over (ƒ)}_(e)+{right arrow over (ƒ)}_(m) and{right arrow over (n)}_(a)={right arrow over (n)}_(e)+{right arrow over(n)}_(m). Then the bias force is given by

$\begin{matrix}{B = {\begin{bmatrix}{\overset{\rightarrow}{\omega} \times H^{T}\overset{\rightarrow}{\omega}} \\{\overset{\rightarrow}{\omega} \times J\overset{\rightarrow}{\omega}}\end{bmatrix} + {\left\lbrack {\begin{matrix}{- {\overset{\rightarrow}{f}}_{e}} \\{- {\overset{\rightarrow}{n}}_{e}}\end{matrix}\begin{matrix}{- {\overset{\rightarrow}{f}}_{m}} \\{- {\overset{\rightarrow}{n}}_{m}}\end{matrix}} \right\rbrack.}}} & (76)\end{matrix}$

The torque on joint i due to acceleration A_(b) can be found by assumingan otherwise stationary manipulator, with all other rates andaccelerations zero. With this assumption, the base frame accelerationcan be expressed in frame i using (70), the frame force required to movethe outboard composite rigid body calculated using (75), and the jointtorque calculated by taking the inner product of this force with φ_(i).This gives the torque on joint i due to base acceleration as thefollowing:

τ_(i) ^(A)=φ_(i) ^(T) I _(i) ^(C)(_(A) T _(b→i))A _(b)  (77)

Let the matrix D be defined as follows:

$\begin{matrix}{D = {\begin{bmatrix}{\varphi_{0}^{T}{I_{0}^{C}\left( {}_{A}T_{b\rightarrow 0} \right)}} \\{\varphi_{1}^{T}{I_{1}^{C}\left( {}_{A}T_{b\rightarrow 1} \right)}} \\\vdots \\{\varphi_{n - 1}^{T}{I_{n - 1}^{C}\left( {}_{A}T_{b\rightarrow{n - 1}} \right)}}\end{bmatrix}.}} & (78)\end{matrix}$

This can be evaluated in code by calculating I_(i) ^(C)φ_(i) as theframe force produced by unit acceleration of joint i with all otherjoints stationary (I_(i) ^(C) is symmetric), then transforming thisforce to the base frame using the identity _(A)T_(i→j) ^(T)=_(F)T_(j→i),∀i,j.

With this definition, the manipulator dynamics equation becomes

τ=M(q){umlaut over (q)}+C(q){dot over (q)}+G(q)+DA _(b) +B.  (79)

Note that, because gravitational G(q) is explicit, A_(b) is the seen(rather than felt) value.

Changing Reference Frames for Matrix D

Because D applies to the base acceleration, it has a frame ofrepresentation, just as the base acceleration does. The formula tochange the frame of expression for D from j to i is given by thefollowing (with _(A)T_(i→j) defined through (68)):

^(i) D= ^(j) D(_(A) T _(i→j)).  (80)

Effect Joint Accelerations on the Base Acceleration on Joint Torque

When the base link is free to move, assuming an otherwise stationarymanipulator, the force that must be exerted on composite rigid body ifor acceleration only by joint i is given by

F_(i)=I_(i) ^(C)φ_(i){umlaut over (q)}_(i)  (81)

Expressing this in the base frame, and changing the sign to representthe effective force applied to the base gives:

^(b)F_(i)=−_(F)T_(i→b)I_(i) ^(C)φ_(i){umlaut over (q)}_(i)  (82)

Summing the contributions of all links and using the identity_(F)T_(i→j) ^(T)=_(A)T_(j→i), ∀i,j gives the following remarkable reuseof the matrix D to calculate the force F_(ma) applied to the base as aresult of manipulator joint accelerations.

F _(ma) =−D ^(T) {umlaut over (q)}.  (83)

Let the total force applied by the manipulator to the base be

F _(m) =F _(ma) +F _(mc) +F _(mg) +F _(me)  (84)

where F_(mc), F_(mg), and F_(me) represent the force due to Coriolis andcentripetal terms, gravity, and external forces, respectively.

If F_(e) represents the external forces applied to the base linkdirectly, then (75) gives

F _(ma) +F _(e) =I _(b) ^(C) A _(b) −F _(mc) −F _(mg) −F _(me)  (85)

where I_(b) ^(C) is the composite rigid-body inertia of the entiremanipulator, including the base.

Substituting in (83) gives

D ^(T) {umlaut over (q)}+I _(b) ^(C) A _(b) =F _(e) +F _(mc) +F _(mg) +F_(me)  (86)

Combining this with (79) gives a new manipulator dynamics equation

$\begin{matrix}{{\begin{bmatrix}I_{b}^{C} & {D(q)}^{T} \\{D(q)} & {M(q)}\end{bmatrix}\begin{bmatrix}A_{b} \\\overset{¨}{q}\end{bmatrix}} = {\begin{bmatrix}{F_{e} + F_{mc} + F_{mg} + F_{me}} \\{\tau - {{C(q)}\overset{.}{q}} - {G(q)} + B}\end{bmatrix}.}} & (87)\end{matrix}$

For N joint degrees of freedom, I_(b) ^(C) is 6×6, D(q) is N×6, and M(q)is N×N. Solving this for A_(b) and {umlaut over (q)} and timeintegrating these gives the free-base composite rigid-body algorithm. Inconcatenating A_(b) and {umlaut over (q)}, A_(b) is placed on top of{umlaut over (q)} to simplify the calculation of the CholeskyDecomposition of the left-hand matrix in (87). Note that this matrix,which must be inverted, is guaranteed not to be singular for a realsystem (otherwise acceleration could be achieved with no force).

Solving for the Accelerations

Let

$\begin{matrix}{{{\overset{\_}{M}(q)} = \begin{bmatrix}I_{b}^{C} & {D(q)}^{T} \\{D(q)} & {M(q)}\end{bmatrix}},} & (88) \\{{\overset{\_}{q} = \begin{bmatrix}A_{b} \\\overset{¨}{q}\end{bmatrix}},{and}} & (89) \\{\overset{\_}{\tau} = {\begin{bmatrix}{F_{e} + F_{mc} + F_{mg} + F_{me}} \\{\tau - {{C(q)}\overset{.}{q}} - {G(q)} + B}\end{bmatrix}.}} & (90)\end{matrix}$

With these definitions, (87) becomes

M (q) q= τ   (91)

an (N+6)-dimensional fully constrained linear equation. CholeskyDecomposition (decomposition into a lower-triangular square root) isideal for solving this for q because M(q), like M(q), is positivedefinite. In this approach, M(q) is decomposed as follows:

M= L L ^(T)  (92)

with L lower triangular.

The constness of I_(b) ^(C) can be exploited in the calculation of L.Let I_(b) ^(C) be decomposed as

I_(b) ^(C)=L_(b)L_(b) ^(T)  (93)

with L_(b) lower triangular. L_(b) is constant and only needs to becalculated once.

Let E be the N×6 matrix satisfying the following:

L_(b)E^(T)=D^(T)  (94)

which can be solved using back substitution with L_(b), and let L_(M),be defined such that

M(q)−EE ^(T) =L _(M) L _(M) ^(T)  (95)

This can be solved using Cholesky decomposition on an N×N matrix.

With these values, L is evaluated as

$\begin{matrix}{{\overset{\_}{L} = \begin{bmatrix}{\overset{\_}{L}}_{b} & 0 \\E & L_{M}\end{bmatrix}},} & (96)\end{matrix}$

Using (73) and (93),

$\begin{matrix}{{{\overset{\_}{L}}_{b} = \begin{bmatrix}{\sqrt{m}I} & 0 \\{{1/\sqrt{m}}H} & {\overset{\_}{L}}_{J}\end{bmatrix}},} & (97)\end{matrix}$

Where L _(J) is lower triangular, and

$\begin{matrix}{{{\overset{\_}{L}}_{J}{\overset{\_}{L}}_{J}^{T}} = {J - {\frac{1}{m}{{HH}^{T}.}}}} & (98)\end{matrix}$

Articulated-Body Simulation Algorithm

The articulated-body algorithm is used for simulation in the softwaretoolkit, running in Order(N) time. More information on such an algorithmcan be found in R. Featherstone, Robot Dynamics Algorithms, KluwerAcademic Publishers, Boston, 1987, and K. W. Lilly, Efficient DynamicSimulation of Robotic Mechanisms, Kluwer Academic Publishers, Boston,1993, both of which are hereby incorporated by reference. This algorithmallows fast simulation of very complex mechanisms.

Articulated Body Inertia

An articulated body is a collection of connected articulating links, asillustrated in FIG. 17. Any single rigid body within the articulatedbody can be used as a handle for defining the relationship between forceand acceleration. The articulated body in FIG. 17 is composed of one ormore rigid links connected by articulated joints. One rigid link, inthis case link i, is taken as the handle to define the relationshipbetween 6′1 frame force (F) and 6′1 frame acceleration (A).

Associated with any handle is a 6×6 articulated-body inertia I^(A) thatsatisfies the following equation for any physically realizable frameacceleration (6×1) A:

F=I ^(A) A−B.  (99)

Here, F is the 6×1 frame force that must be applied to the articulatedbody to achieve 6×1 frame acceleration A. (For vector force {right arrowover (ƒ)}, moment {right arrow over (n)}, linear acceleration {rightarrow over (a)}, and angular acceleration {right arrow over (α)},represented as 3×1 column vectors, F=[{right arrow over (ƒ)}^(T){rightarrow over (n)}^(T)]^(T) and A=[{right arrow over (a)}^(T){right arrowover (α)}^(T)]^(T).) The 6×1 frame force B is a bias force that is afunction of external forces on the links, internal forces on the links,gravity, and link velocities. B represents all contributors to the forceon the link except acceleration A. It is the force required to produceno acceleration of the handle.

The iterative formulas to calculate I^(A) and B are given by thefollowing:

$\begin{matrix}{{I_{k}^{A} = \left( {I_{j}^{A} + I_{i}^{A} - {\frac{1}{\varphi_{j}^{T}I_{j}^{A}\varphi_{j}}I_{j}^{A}\varphi_{j}\varphi_{j}^{T}I_{j}^{A}}} \right)},} & (100) \\{B_{k} = {{\frac{1}{\varphi_{j}^{T}I_{j}^{A}\varphi_{j}}\left( {\tau_{j} - {\varphi_{j}^{T}B_{j}}} \right)I_{j}^{A}\varphi_{j}} + B_{j} + {B_{i}.}}} & (101)\end{matrix}$

When the base is free, these equations can be continued to the baselink. Then, at the base link, the frame acceleration will be given by

A _(b)=(I _(b) ^(A))⁻¹(F _(b) −B _(b)).  (102)

Here, A_(b) is the acceleration that is felt, rather than seen. If theframe is stationary with respect to a gravitational field, then A_(b)will show acceleration upward.

Free-Base Dynamics Simulation

The Articulated-Body Inertia Algorithm and the Composite Rigid-BodyInertia Algorithm were modified for floating-base manipulators asdescribed above. Testing was done by 1) verifying the algorithmsconserve energy and momentum and 2) comparing the two independentalgorithms for equality of results. FIGS. 18A-18D show side-by-sidecomparisons of the two algorithms simulating a free-base three-linkmechanism.

The teachings of U.S. Pat. No. 6,757,587, titled “METHODS AND APPARATUSFOR DYNAMICALLY REPROGRAMMING REMOTE AUTONOMOUS AGENTS” and issued onJun. 29, 2004, are hereby incorporated by reference.

One skilled in the art will appreciate further features and advantagesbased on the above-described embodiments. Accordingly, the claims arenot to be limited by what has been particularly shown and described byway of example. The appended claims are incorporated by reference hereinand are considered to represent part of the disclosure and detaileddescription of this patent application.

All publications and references cited herein are expressly incorporatedherein by reference in their entirety.

1. A method of controlling a robotic apparatus comprising a plurality ofmovable elements coupled to one another via one or more joints definingone or more degrees of freedom of movement for each of the movableelements, at least one of said movable elements comprising a mobile basefor the apparatus, the control method comprising: receiving inputsignals indicating a desired position or motion constraint for at leastone of the movable elements; generating, based on the input signals, aplurality of control signals to control movement of the mobile base andat least one other movable element to achieve the desired motion byoptimizing a pre-defined scalar function of the motion and positions ofthe movable elements and the mobile base; applying the control signalsto the mobile base and the at least one other movable element toconcurrently move the mobile base and the at least one other movableelement to effect the desired motion.
 2. The method of claim 1, whereinat least one of the movable elements comprises an end effector.
 3. Themethod of claim 2, wherein the end effector comprises any of a gripper,a cutter, a saw, a haptic element, a multi-fingered end effector.
 4. Themethod of claim 1, wherein at least one of the movable elements providescontact with the environment for the purpose of locomotion.
 5. Themethod of claim 1, wherein the mobile base comprises any of a wheel, atrack, and a rail.
 6. The method of claim 1, wherein the mobile basecomprises any of a digging, climbing, and spelunking apparatus.
 7. Themethod of claim 1, wherein the mobile base comprises any of a boat,submarine, and hovercraft.
 8. The method of claim 1, wherein the mobilebase comprises any of an airplane, helicopter, and lighter-than-airvehicle.
 9. The method of claim 1, wherein the plurality of movableelements comprise a robotic arm and a mobile base.
 10. The method ofclaim 1, wherein the input signals are generated by a user input device.11. The method of claim 10, further comprising, (i) displaying aplurality of views of the robotic apparatus pose via to a user, and (ii)allowing the user to control the robotic apparatus from each of theplurality of the views with the user input device.
 12. The method ofclaim 11, wherein the user input device requires only a single useraction to switch control between the plurality of the views.
 13. Themethod of claim 11, wherein the plurality of views comprise a pluralityof rendered views.
 14. The method of claim 10, further comprising, (i)displaying a plurality of views of the robotic apparatus pose via to auser, and (ii) allowing the user to control the robotic apparatus fromeach of the plurality of views with the user input device, and (iii)automatically updating each of the plurality of views based on a user'scontrol input in any one view.
 15. The method of claim 1, wherein theinput signals are transmitted over any of audio signal, electromagneticsignal, or a computer network.
 16. The method of claim 1, wherein themobile base has two or more degrees of freedom of movement comprisinglinear velocity and angular velocity.
 17. The method of claim 1,comprising applying the control signals to one or more actuators coupledto the one or more movable elements to effect the desired motion. 18.The method of claim 1, wherein the scalar function is associated withmovement of the robotic apparatus, a gradient of potential energyassociated with movement of the robotic apparatus, singularityavoidance, accuracy, joint-limit avoidance, obstacle avoidance.
 19. Themethod of claim 1, wherein the scalar function signifies a constraint onmovement of at least one movable element.
 20. The method of claim 1,wherein the movable elements of the robotic apparatus exhibit,collectively, at least seven degrees of freedom of movement.
 21. Themethod of claim 1, wherein the scalar function is defined by:$\mu = {{\frac{1}{2}{\overset{.}{q}}^{T}W\overset{.}{q}} + {\alpha \; {\overset{.}{q}}^{T}{F.}}}$where q is a n-length vector representation of the positions of the oneor more joints and the mobile base; with {dot over (q)} being its timederivative (change in position over time); W is a matrix function W(q);F is a vector function of configuration; and α is a scalar function ofconfiguration.
 22. The method of claim 21, wherein W, F and α areselected to define a control strategy for controlling the roboticapparatus.
 23. The method of claim 1, wherein generating the controlsignals comprises determining control vectors using the followingequation:$\overset{.}{q} = {\left\lbrack \frac{J}{N_{J}^{T}W} \right\rbrack^{- 1}\left\lbrack \frac{V}{{- \alpha}\; N_{J}^{T}F} \right\rbrack}$where q is a n-length vector representation of the positions of the oneor more joints and the mobile base; with {dot over (q)} being its timederivative (change in position over time); V is a m-length vectorrepresentation of desired motion of at least one of the plurality ofmovable elements; J is an (m×n) manipulator Jacobian; N_(j) ^(T) is ann×(n−m) set of vectors defined such that JN_(J) ^(T)=0; W is a matrixfunction of q; F is a vector function of q; and αis a scalar function ofq.
 24. A robotic system, comprising: a plurality of movable elementscoupled to one another via one or more joints defining one or moredegrees of freedom of movement for each of the movable elements, atleast one of said movable elements comprising a mobile base; a userinput device for receiving input signals indicating a desired positionor motion constraint for at least one point on one of the movableelements; one or more processors coupled to the input device andgenerating a plurality of control signals to control movement of themobile base and at least one other movable element to achieve thedesired motion by optimizing a pre-defined scalar function of motion andpositions of the movable elements and the mobile base; one or moreactuators coupled to the plurality of movable elements and to the one ormore processors, the one or more actuators effecting the desired motionby concurrently moving the mobile base and the at least one othermovable element in accordance with the control signals.
 25. The systemof claim 24, wherein at least one of the movable elements comprises anend effector, the end effector comprising any of a gripper, a cutter, asaw, a haptic element, a multi-fingered end effector.
 26. The system ofclaim 24, wherein the base comprises any of a wheel, a track, and arail.
 27. The system of claim 24, wherein the plurality of movableelements comprise a robotic arm and a mobile base.
 28. The system ofclaim 24, wherein scalar function is defined by:$\mu = {{\frac{1}{2}{\overset{.}{q}}^{T}W\overset{.}{q}} + {\alpha \; {\overset{.}{q}}^{T}{F.}}}$where q is a n-length vector representation of the positions of the oneor more joints and the mobile base; with {dot over (q)} being its timederivative (change in position over time); W is a matrix function W(q);F is a vector function of configuration; and α is a scalar function ofconfiguration.
 29. The system of claim 24, further comprising anoperator control unit that (i) displays a plurality of views of therobotic apparatus pose, and (ii) allows a user to control the roboticapparatus from each of the plurality of views.
 30. The system of claim29, wherein the operator control unit requires only a single user actionto switch control between the plurality of views.
 31. The system ofclaim 29, wherein the plurality of views comprise a plurality ofrendered views.
 32. The system of claim 24, further comprising anoperator control unit that (i) displays a plurality of views of therobotic apparatus pose, (ii) allows a user to control the roboticapparatus from each of the plurality of views, and (iii) automaticallyupdates each of the plurality of views based on a user's control inputin any one view.
 33. A robotic system, comprising: a plurality ofmovable elements coupled to one another via one or more joints definingone or more degrees of freedom of movement for each of the movableelements, at least one of said movable elements comprising a mobilebase; a user input device for receiving input signals indicating adesired motion for at least one point on one of the movable elements;one or more processors receiving the input signals and generating aplurality of control signals to control movement of the mobile base andat least one other movable element to achieve the desired motion; one ormore actuators coupled to the plurality of movable elements and to theone or more processors, the one or more actuators effecting the desiredmotion by concurrently moving the mobile base and the at least one othermovable element in accordance with the control signals, wherein the oneor more processors generates the control signals using the followingequation:$\overset{.}{q} = {\left\lbrack \frac{J}{N_{J}^{T}W} \right\rbrack^{- 1}\left\lbrack \frac{V}{{- \alpha}\; N_{J}^{T}F} \right\rbrack}$where q is a n-length vector representation of the positions of the oneor more joints and the mobile base; with {dot over (q)} being its timederivative; V is a m-length vector representation of desired motion ofat least one point on one of the plurality of movable elements; J is an(m×n) manipulator Jacobian; N_(j) ^(T) is an n×(n−m) set of vectorsdefined such that JN_(J) ^(T)=0; W is a matrix function W(q); F is thegradient of scalar function f(q); α is a scalar.
 34. The system of claim33, wherein W, F and α are user-selected to define a control strategyfor controlling the robotic apparatus.
 35. The system of claim 34,wherein the one or more processors generate the control signals so as tooptimize a merit function defined as:$\mu = {{\frac{1}{2}{\overset{.}{q}}^{T}W\overset{.}{q}} + \; {{\overset{.}{q}}^{T}F\; \alpha}}$where μ represents a parameter to be optimized that reflects the controlstrategy.
 36. The system of claim 33, further comprising an operatorcontrol unit that (i) displays a plurality of views of the roboticapparatus pose, and (ii) allows a user to control the robotic apparatusfrom each of the plurality of the views.
 37. The system of claim 36,wherein the operator control unit requires only a single user action toswitch control between the plurality of views.
 38. The system of claim36, wherein the plurality of views comprise a plurality of renderedviews.
 39. The system of claim 33, further comprising an operatorcontrol unit that (i) displays a plurality of views of the roboticapparatus pose, (ii) allows a user to control the robotic apparatus fromeach of the plurality of views, and (iii) automatically updates each ofthe plurality of views based on a user's control input in any one view.